One the most general and well accepted means of describing the spectral and spatial reflective scattering properties of a material is by use of the bidirectional reflectance distribution function (BRDF). The BRDF is a fundamental description of the appearance of the surface of a material, and many other appearance attributes (such as, gloss, haze, and color) can be represented in terms of integrals of the BRDF over specific geometries and spectral conditions. Specification of the BRDF is critical to the marketability of consumer products, such as, automobiles, cosmetics and electronics. The microstructure associated with a material affects the BRDF and specific properties can often be inferred from measurement of the BRDF. The angular distribution of reflectively scattered light described by the BRDF can be used to render the appearance of materials or to predict the color appearance under varying geometrical conditions. The quality of the rendering or color prediction depends heavily on the accuracy of the BRDF of the materials being rendered.
Gonioapparent objects or materials exhibit the characteristic of changing their appearance with change in illumination angle or viewing angle. Automotive finishes (paints) containing metallic flake pigments or special effect flake pigments, such as, pearlescent flake pigments are examples of gonioapparent materials. Unlike solid colors which can be characterized at a single measurement geometry, gonioapparent colors require measurements under a variety of illumination and viewing geometries to describe their color appearance characteristics. Finishes containing metallic flakes are generally characterized by making three color measurements at different aspecular angles. ASTM standard E-2194, which is hereby incorporated by reference, describes a standard practice for multi-angle color measurement of metal flake pigmented materials. Finishes containing special effect flake pigments that are hue shifting materials, such as, pearlescent pigments, also must be measured at multiple geometries which vary in both aspecular angle and illumination angle to characterize their color behavior.
In order to render objects on a video screen, or print media, or otherwise predict the color appearance of an object at a given illumination and viewing geometry, the object's color at many thousands of combinations of illumination and viewing angles must be calculated.
There are three basic techniques that have been used for the task of calculating all the required combinations of illumination and view.                1) The first technique is to actually measure the color of the object at several thousand combinations of illumination and view with an instrument such as a goniospectrophotometer, or goniocolorimeter. This requires that a sufficient number of measurements be made so that interpolation of the data to predict the color of the object at intermediate geometries can be done with sufficient accuracy. However, instruments with the required geometric flexibility and photometric accuracy are costly and very slow. Complete characterization of a single color requires several hours of measurement time using this technique.        2) A second technique is to develop a physical model of the finish (color) and then use a technique, such as, radiative transfer theory to calculate the color at all of the required angular combinations. While techniques of this type can be used to produce visually pleasing renderings, development and tuning of the model to match the behavior of a physical standard is extremely difficult and time consuming and may in fact be impossible to do with sufficient fidelity.        3) The third technique is a combination of the first two with the advantage of requiring far fewer measurements than the first technique and a far less rigorous model of the finish than the second technique. This third technique involves making a limited number (typically 3-5) of color measurements of the object to be rendered and then modeling the interpolation of this measured data to the required angular combinations. This technique can utilize 3-angle measurement data already contained in databases typically used to store color characteristics of gonioapparent materials. The models used to extrapolate this data to other angular combinations do not require individual tuning and are based on simple physical parameters of the surface of the material.        
For rendering or color prediction applications requiring measurement of a vast array of colors, which match actual physical standards and are not just “realistic looking” synthetic colors, the combination technique as described above is the preferred solution.
Alman (U.S. Pat. No. 4,479,718) led to the eventual wide spread adoption of a three aspecular angle measurement system for characterization of finishes containing metal flake pigments in combination with absorbing and or scattering pigments. This measurement system serves as the basis for such international standards as ASTM E-2194 and DIN 6175-2. In practice, this characterization approach also works well for formulation and control of finishes containing hue shifting (pearlescent) pigments once pigmentation has been established.
While the concept of describing the gonioapparent color behavior of a material by measurements made at three aspecular angles is useful for formulation and control, and can be used to predict if a pair of samples will match under various measurement or viewing geometries, it is not well suited to predict the absolute color of a material as the measurement and viewing geometries change. For instance, while the same general color change predicted by aspecular measurements hold as the illumination angle is changed, the magnitude of the color change is not well predicted. FIG. 2 shows a plot of tristimulus value Y as a function aspecular angle for a variety of illumination angles for an automotive paint specimen containing metal flake pigment. While there is a trend to increasing value of Y as the aspecular angle decreases, there are large differences in the absolute value of Y at a given aspecular angle as the illumination angle is changed.
A method is needed to predict the absolute color of a specimen, under any measurement or viewing geometry, from a limited (<10) set of color measurements.